Finally, we note that Theorem REF  is related to the problem of geometrically characterizing removable sets for Lipschitz harmonic functions (RLH sets) in \(\mathbb {H}\) . The definition of an RLH set in \(\mathbb {H}\)  is completely analogous to its Euclidean counterpart, except that, in \(\mathbb {H}\) , a function is called harmonic if it is a solution to the sub-Laplacian equation \(\Delta _{\mathbb {H}}u=0\) . RLH sets in Heisenberg groups were introduced in [1]} and it was shown there that if \(E \subset \mathbb {H}\)  is compact, then it is RLH if \(\mathcal {H}^3(E)=0\) , while it is not RLH if \(\dim _H (E)>3\) . Moreover, totally disconnected RLH sets with positive 3-dimensional Hausdorff measure were produced in [1]}, [3]}. On the other hand, it was proved in [4]} that if \(\mu \)  is a non-trivial compactly supported Radon measure in \(\mathbb {H}\)  with 3-upper growth, such that \(T^\mathsf {R}\)  is bounded in \(L_2(\mu )\)  then \(\operatorname{spt}\mu \)  is not RLH. An analogous result holds in \(\mathbb {R}^n\) , see [5]}, and combined with the \(L_2\) –boundedness of Riesz transforms on Lipschitz graphs implies that compact subsets of 1-codimensional Lipschitz graphs with positive \((n-1)\) -Hausdorff measure are not RLH. This can be used to show that if a compact set \(E \subset \mathbb {R}^n\)  with \(H^{n-1}(E)<\infty \)  is RLH then it is purely \((n-1)\) –unrectifiable. To our knowledge, this is the only known proof for this implication.
